Find $p$-Sylow subgroups of $\Bbb Ζ_n$.

Question

Find $p$-Sylow subgroups of $\Bbb Ζ_n$.

Ι found that $n$ has $2$ primes $p_1$, $p_2$ as factors. We have $1$ $p_1$-Sylow and $1$ $p_2$-Sylow subgroups because $\Bbb Z_n$ is cyclic and abelian.

Now I have to determine what are the subgroups exactly but I don't know how to do it.

Answer 1

If $n=p_1^m\cdot p_2^n$, you should try the subgroups $H, K$ of orders $p_1^m$ and $p_2^n$, respectively. To be more precise, $H=\{ z\in\mathbb{Z}_n\ |\ z=ap_2^n, a\in\mathbb{Z}\}$. You can check that this does indeed give you a subgroup, and that the order of this is $p_1^m$. The latter is seen by first observing that $p_1^m\cdot z = a\cdot p_1^m\cdot p_2^n = a\cdot n = 0$, and then by, choosing $a=1$, you will see that the minimal integer killing all the elements is precisely $p_1^m$. For $K$, you just replace $p_2^n$ with $p_1^m$ and vice versa.

Answer 2

If $n=p^mq^l$, then the Sylow subgroups are $\Bbb Z_{p^m}$ and $\Bbb Z_{q^l}$.